If $\sin x = \frac{3}{5}$ and $\cos y = -\frac{12}{13}$,where $x$ and $y$ both lie in the second quadrant,find the value of $\sin (x+y)$.

For $\alpha, \beta \in \left(0, \frac{\pi}{2}\right)$,let $3 \sin (\alpha+\beta)=2 \sin (\alpha-\beta)$ and a real number $k$ be such that $\tan \alpha=k \tan \beta$. Then the value of $k$ is equal to :

If $\alpha$ is in the $3^{\text{rd}}$ quadrant,$\beta$ is in the $2^{\text{nd}}$ quadrant such that $\tan \alpha = \frac{1}{7}$ and $\sin \beta = \frac{1}{\sqrt{10}}$,then find the value of $\sin(2\alpha + \beta)$.

If $\theta+\phi=\frac{\pi}{4}$, then $(1+\tan \theta)(1+\tan \phi)$ is equal to

If $\cot \alpha = \frac{1}{2}$ and $\sec \beta = -\frac{5}{3}$,where $\alpha \in \left(\pi, \frac{3\pi}{2}\right)$ and $\beta \in \left(\frac{\pi}{2}, \pi\right)$,then the value of $\tan(\alpha + \beta)$ is:

If $2 \sin \left(\theta+\frac{\pi}{3}\right)=\cos \left(\theta-\frac{\pi}{6}\right)$,then $\tan \theta=$